Locally repairable codes (LRCs) with $(r,\delta)$ locality were introduced by Prakash \emph{et al.} into distributed storage systems (DSSs) due to their benefit of locally repairing at least $\delta-1$ erasures via other $r$ survival nodes among the same local group. An LRC achieving the $(r,\delta)$ Singleton-type bound is called an optimal $(r,\delta)$ LRC. Constructions of optimal $(r,\delta)$ LRCs with longer code length and determining the maximal code length have been an important research direction in coding theory in recent years. In this paper, we conduct further research on the improvement of maximum code length of optimal $(r,\delta)$ LRCs. For $2\delta+1\leq d\leq 2\delta+2$, our upper bounds largely improve the ones by Cai \emph{et al.}, which are tight in some special cases. Moreover, we generalize the results of Chen \emph{et al.} and obtain a complete characterization of optimal $(r=2, \delta)$-LRCs in the sense of geometrical existence in the finite projective plane $PG(2,q)$. Within this geometrical characterization, we construct a class of optimal $(r,\delta)$ LRCs based on the sunflower structure. Both the construction and upper bounds are better than previous ones.
翻译:Prakash \ emph{et al.} 在分布式储存系统(DSS)中引入了(r,\ delta) 本地可修理代码(LRCs), 其值为$(r,\ delta) 美元 。 Prakash 代码长度较长且确定最大代码长度的LRCs 建造是近年来编码理论的一个重要研究方向。 在本文中,我们进一步研究了如何通过其他当地组群中的其他美元生存节点对至少$(r,\delta) 美元进行本地修理。 对于2\ delta+1\leq dleq 2\delta+2美元来说, 我们的顶端框大大改进了Cai \ emph{etelta $(r,\ delta) 美元 (r, delta) $(r, delph) 美元(r, leqr) 和 al., 在某些特殊案例中,我们一般地平面的 RC 结构中, $ 的平面结构中, ial=L2, roqr= deal deal deal ruals) exal exal exal supertial exal sual superal ex 。